A characterization of classical Riemannian manifolds by curvature operators (Q2474141)

The authors relate algebraic properties of certain natural operators in Riemannian geometry to the underlying geometry of the manifold. Let \((M,g)\) be an n-dimensional Riemannian manifold. The classical Jacobi curvature operator \(J_x:M_p\to M_p\) induced by a unit vector \(x\in M_p\), is defined by \(J_x(u)=R(u,x,x)\), where \(R\) is the curvature tensor. In 1990 G. Stanilov defined the skew-symmetric curvature operator \(k_{E^2}:M_p\to M_p\) and the generalized Jacobi operator \(S_{E^m}:M_p\to M_p\). Namely, let \(x,y\) is an orthonormal basis of a 2-dimensional tangent plane \(E^2\subset M_p\), then \(k_{E^2}(u)=R(x,y,u)\). Now let \(E^m\) be an m-dimensional tangent subspace with an orthonormal basis \((e_i)\). Thus \[ S_{E^m}(u)=\sum_{i=1}^m J_{e_i}(u)\,. \] The main theorem of the paper is the following: Theorem. Let \(\dim M=4\). Then \((M,g)\) is conformally flat or an Einstein manifold if and only if for every tangent plane \(E^2\) the operators \(k_{E^2}\) and \(S_{E^2}\) are orthogonal to each other on the orthogonal component \(^\perp E^2\) of \(E^2\).